Average Gradient & Average Rate of Change (Grade 12)

The gradient of the straight line joining two points on a curve, why it is an average rather than the gradient at a point, and how it becomes the derivative as the two points close together.

A curve does not have one gradient — it changes at every point. The average gradient between two points is the gradient of the straight line joining them.

1The formula

average gradient = f(b) − f(a)b − a = change in ychange in x

It is the ordinary gradient formula from Grade 10, with the y-values read off the function.

The average gradient of f(x) = x² + 1 between x = 1 and x = 3 is the gradient of the straight line through (1 ; 2) and (3 ; 10).

Worked ExampleDetermine the average gradient of f(x) = x² + 1 between x = 1 and x = 3

1
f(1) = 1² + 1 = 2 and f(3) = 3² + 1 = 10.
Find both y-values first.
2
Average gradient = 10 − 23 − 1.
3
= 82 = 4.
Check on the sketch: the line through (1 ; 2) and (3 ; 10) rises 8 across 2. ✓

⚠️

The average gradient is not the gradient at x = 1 or at x = 3. It is the gradient of the line between them — the curve is steeper at 3 and shallower at 1.

2Why it is called an average

Between x = 1 and x = 3 the actual gradient of the curve climbs from 2 to 6. The value 4 sits between them — it is the single straight-line gradient that would take you from the first point to the second.

3The link to the derivative

f′(x) = lim_h→0 f(x + h) − f(x)h

That is the average gradient between x and x + h. As h shrinks to nothing the two points merge, the straight line becomes the tangent, and the average gradient becomes the gradient at a point — the derivative.

Worked ExampleShow that the average gradient of f(x) = x² + 1 near x = 2 approaches 4

1
Between x = 2 and x = 2,001: f(2,001) − f(2)0,001.
2
= 5,004001 − 50,001 = 4,001.
3
As h → 0 this tends to 4, and f′(x) = 2x gives f′(2) = 4. ✓
The average gradient becomes the derivative.

💡

In a rate-of-change question the same formula answers 'how fast on average'. Average speed over a journey is exactly the average gradient of the distance–time graph.

Practice exercises

Work each one out, then click to reveal the answer.

📄 Worksheet ✅ Memo 📝 Exam paper 🗝️ Marking guide

1
Determine the average gradient of f(x) = x² + 1 between x = 1 and x = 3.
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10 − 23 − 1 = 4
2
Determine the average gradient of f(x) = x² + 1 between x = −1 and x = 2.
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5 − 22 + 1 = 1
3
Write down the formula for average gradient between x = a and x = b.
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f(b) − f(a)b − a
4
Is the average gradient the same as the gradient at a point? Explain.
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No — it is the gradient of the straight line joining two points; the gradient at a point is the gradient of the tangent there.
5
Determine the average gradient of f(x) = x² between x = 2 and x = 2,001 (three decimals).
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4,001… which tends to 4 as the interval shrinks
6
What does the average gradient become as the two points move together?
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The derivative — the gradient of the tangent at that point.
7
A car travels 240 km in 3 hours. What is its average rate of change of distance?
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2403 = 80 km/h
8
Determine the average gradient of f(x) = 2x + 5 between any two points.
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2 — for a straight line the average gradient is the same everywhere.